Understanding the Age Word Problem

This problem involves the ages of two people, Savan and Akshan, at two different points in time: currently and 10 years from now. We are given two relationships between their ages at these times and need to find Savan's current age.

Setting up Equations for Savan's and Akshan's Ages

To solve this type of age problem, we can use variables to represent the unknown ages and translate the given information into algebraic equations.

• Let Savan's current age be \(S\) years.
• Let Akshan's current age be \(A\) years.

Now, let's express the given relationships as equations:

1. Current Age Relationship: Savan's current age is four times the age of Akshan.
   This translates to: \(S = 4A\) (Equation 1)
2. Future Age Relationship (10 years from now):
   • In 10 years, Savan's age will be \(S + 10\).
   • In 10 years, Akshan's age will be \(A + 10\).
   Savan’s age will be twice the age of Akshan 10 years from now.
   This translates to: \(S + 10 = 2 \times (A + 10)\) (Equation 2)

Solving the Equations to Find Ages

We now have a system of two linear equations with two variables:

Equation 1: \(S = 4A\)

Equation 2: \(S + 10 = 2(A + 10)\)

We can use the substitution method to solve this system. Substitute the expression for \(S\) from Equation 1 into Equation 2:

Replace \(S\) with \(4A\) in Equation 2:

\(4A + 10 = 2(A + 10)\)

Now, solve for \(A\):

• Distribute the 2 on the right side:
  \(4A + 10 = 2A + 20\)
• Subtract \(2A\) from both sides of the equation:
  \(4A - 2A + 10 = 2A - 2A + 20\)
  \(2A + 10 = 20\)
• Subtract 10 from both sides:
  \(2A + 10 - 10 = 20 - 10\)
  \(2A = 10\)
• Divide by 2:
  \(\frac{2A}{2} = \frac{10}{2}\)
  \(A = 5\)

So, Akshan's current age is 5 years.

Now that we have Akshan's current age (\(A=5\)), we can find Savan's current age (\(S\)) using Equation 1:

\(S = 4A\)

\(S = 4 \times 5\)

\(S = 20\)

Thus, Savan's current age is 20 years.

Verification

Let's check if these ages satisfy the conditions given in the problem:

• Currently: Savan is 20 years old, Akshan is 5 years old. Is 20 four times 5? Yes, \(20 = 4 \times 5\). The first condition is met.
• 10 years from now:
  • Savan's age will be \(20 + 10 = 30\) years.
  • Akshan's age will be \(5 + 10 = 15\) years.
  Is Savan's age (30) twice Akshan's age (15)? Yes, \(30 = 2 \times 15\). The second condition is met.

Both conditions are satisfied, confirming our solution is correct.

Savan’s current age is 20 years.

Revision Table: Key Concepts

Additional Information: Solving Age Word Problems

Age word problems are common in algebra. They often involve relationships between people's ages at different points in time (past, present, future). Here are some tips for solving them:

• Identify the unknowns: Determine whose age you need to find, usually at the present time. Assign variables.
• Represent ages at different times: If the problem mentions ages in the past or future, express these ages in terms of the current age variable. For example, \(x\) years ago, a person's age would be \(Current\_Age - x\); \(y\) years from now, their age would be \(Current\_Age + y\).
• Set up equations: Carefully read the problem to find the relationships between the ages at the given times. Translate these relationships into algebraic equations. Each distinct relationship usually gives you one equation.
• Solve the system of equations: Use methods like substitution or elimination to solve for the unknown variables.
• Answer the question asked: Make sure you answer the specific question (e.g., find Savan's *current* age, not Akshan's age or their age in 10 years).
• Check your solution: Plug the values you found back into the original word problem description to ensure they make sense and satisfy all conditions. This helps catch calculation errors.

Practice with different variations of age problems will help you become more comfortable setting up and solving the equations.